Your chickens don't have a cold, they don't get them. They either have something like mycoplasma or coryza that is going to need to be treated with antibiotics to clear up or they have a viral infection like infectious bronchitis, which will pass on its own or will kill them, you can't do anything for viruses, just like you can't do anything to cure a human cold that's caused by a virus.
Depending on what they have, they will possibly carry it for life and it will spring back up again whenever they are stressed (such as when coming into lay, during a cold snap, etc).
As to whether you can eat the eggs, I don't see why not. None of those diseases are transmittable to humans. If you do end up having to treat with antibiotics then you wouldn't be able to eat the eggs until the end of the withdrawal period.
